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Post code that demonstrates the use of the interact command in Sage here.    It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email sage-support@googlegroups.com. This is a collection of pages demonstrating the use of the **interact** command in Sage.
It should be easy to just scroll through and copy/paste examples into Sage notebooks.
If you have suggestions on how to improve interact, add them [[interact/Suggestions|here]]
or email the
sage-support mailing list. Of course, your own examples are also welcome!
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 * [:interact/graph_theory:Graph Theory]
 * [:interact/calculus:Calculus]
 * [:interact/diffeq:Differential Equations]
 * [:interact/linear_algebra:Linear Algebra]
 * [:interact/algebra:Algebra]
 * [:interact/number_theory:Number Theory]
 * [:interact/web:Web Applications]
 * [:interact/bio:Bioinformatics]
Documentation links:
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== Miscellaneous Graphics ==   * [[http://doc.sagemath.org/html/en/reference/repl/sage/repl/ipython_kernel/interact.html| interacts in the Jupyter notebook]] (see this page and the two following ones)
  * [[https://github.com/sagemath/sagenb/blob/master/sagenb/notebook/interact.py|interacts in the legacy SageNB notebook]] (many helpful examples)
  * [[https://github.com/sagemath/sagecell/blob/master/interact_compatibility.py|Sage Cell Server implementation]]
  * [[https://github.com/sagemathinc/cocalc/blob/master/src/smc_sagews/smc_sagews/sage_salvus.py#L348|CoCalc Sage worksheet implementation]]
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=== Catalog of 3D Parametric Plots ===
{{{
var('u,v')
plots = ['Two Interlinked Tori', 'Star of David', 'Double Heart',
         'Heart', 'Green bowtie', "Boy's Surface", "Maeder's Owl",
         'Cross cap']
plots.sort()
Examples:

 * [[interact/algebra|Algebra]]
 * [[interact/bio|Bioinformatics]]
 * [[interact/calculus|Calculus]]
 * [[interact/complex|Complex Analysis]]
 * [[interact/cryptography|Cryptography]]
 * [[interact/diffeq|Differential Equations]]
 * [[interact/graphics|Drawing Graphics]]
 * [[interact/dynsys|Dynamical Systems]]
 * [[interact/fractal|Fractals]]
 * [[interact/games|Games and Diversions]]
 * [[interact/geometry|Geometry]]
 * [[interact/graph_theory|Graph Theory]]
 * [[interact/groups|Groups]]
 * [[interact/linear_algebra|Linear Algebra]]
 * [[interact/Loop Quantum Gravity|Loop Quantum Gravity]]
 * [[interact/misc|Miscellaneous]]
 * [[interact/number_theory|Number Theory]]
 * [[interact/stats|Statistics/Probability]]
 * [[interact/topology|Topology]]
 * [[interact/web|Web Applications]]

== Explanatory example: Taylor Series ==

This is the code and a mockup animation of the interact command. It defines a slider, seen on top, that can be dragged. Once dragged, it changes the value of the variable "order" and the whole block of code gets evaluated. This principle can be seen in various examples presented on the pages above!

{{{#!sagecell
x = SR.var('x')
x0 = 0
f = sin(x) * e^(-x)
p = plot(f, -1, 5, thickness=2)
dot = point((x0, f(x=x0)), pointsize=80, rgbcolor=(1, 0, 0))
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def _(example=selector(plots, buttons=True, nrows=2),
      tachyon=("Raytrace", False), frame = ('Frame', False),
      opacity=(1,(0.1,1))):
    url = ''
    if example == 'Two Interlinked Tori':
        f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*cos(u), 4+sin(v))
        f2 = (8+(3+cos(v))*cos(u), 3+sin(v), 4+(3+cos(v))*sin(u))
        p1 = parametric_plot3d(f1, (u,0,2*pi), (v,0,2*pi), color="red", opacity=opacity)
        p2 = parametric_plot3d(f2, (u,0,2*pi), (v,0,2*pi), color="blue",opacity=opacity)
        P = p1 + p2
    elif example == 'Star of David':
        f_x = cos(u)*cos(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        f_y = cos(u)*sin(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        f_z = sin(u)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, 0, 2*pi),opacity=opacity)
    elif example == 'Double Heart':
        f_x = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) + abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*sin(v)
        f_y = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) - abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*cos(v)
        f_z = sin(u)*(abs(cos(4*u/4))^1 + abs(sin(4*u/4))^1)^(-1/1)
        P = parametric_plot3d([f_x, f_y, f_z], (u, 0, pi), (v, -pi, pi),opacity=opacity)
    elif example == 'Heart':
        f_x = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
        f_y = sin(u) *(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
        f_z = v
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -1, 1), frame=False, color="red",opacity=opacity)
    elif example == 'Green bowtie':
        f_x = sin(u) / (sqrt(2) + sin(v))
        f_y = sin(u) / (sqrt(2) + cos(v))
        f_z = cos(u) / (1 + sqrt(2))
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -pi, pi), frame=False, color="green",opacity=opacity)
    elif example == "Boy's Surface":
        url = "http://en.wikipedia.org/wiki/Boy's_surface"
        fx = 2/3* (cos(u)* cos(2*v) + sqrt(2)* sin(u)* cos(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        fy = 2/3* (cos(u)* sin(2*v) - sqrt(2)* sin(u)* sin(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        fz = sqrt(2)* cos(u)* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, pi), plot_points = [90,90], frame=False, color="orange",opacity=opacity)
    elif example == "Maeder's Owl":
        fx = v *cos(u) - 0.5* v^2 * cos(2* u)
        fy = -v *sin(u) - 0.5* v^2 * sin(2* u)
        fz = 4 *v^1.5 * cos(3 *u / 2) / 3
        P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, 1),plot_points = [90,90], frame=False, color="purple",opacity=opacity)
    elif example =='Cross cap':
        url = 'http://en.wikipedia.org/wiki/Cross-cap'
        fx = (1+cos(v))*cos(u)
        fy = (1+cos(v))*sin(u)
        fz = -tanh((2/3)*(u-pi))*sin(v)
        P = parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red",opacity=opacity)
    else:
        print "Bug selecting plot?"
        return


    html('<h2>%s</h2>'%example)
    if url:
        html('<h3><a target="_new" href="%s">%s</a></h3>'%(url,url))
    show(P, viewer='tachyon' if tachyon else 'jmol', frame=frame)
def _(order=slider([1 .. 12])):
  ft = f.taylor(x, x0, order)
  pt = plot(ft, -1, 5, color='green', thickness=2)
  pretty_print(html(r'$f(x)\;=\;%s$' % latex(f)))
  pretty_print(html(r'$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$' % (x0, latex(ft), order+1)))
  show(dot + p + pt, ymin=-.5, ymax=1)
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attachment:parametricplot3d.png

=== Interactive rotatable raytracing with Tachyon3d ===

{{{
C = cube(color=['red', 'green', 'blue'], aspect_ratio=[1,1,1],
         viewer='tachyon') + sphere((1,0,0),0.2)
@interact
def example(theta=(0,2*pi), phi=(0,2*pi), zoom=(1,(1,4))):
    show(C.rotate((0,0,1), theta).rotate((0,1,0),phi), zoom=zoom)
}}}


attachment:tachyonrotate.png


=== Interactive 3d plotting ===
{{{
var('x,y')
@interact
def example(clr=Color('orange'), f=4*x*exp(-x^2-y^2), xrange='(-2, 2)', yrange='(-2,2)',
    zrot=(0,pi), xrot=(0,pi), zoom=(1,(1/2,3)), square_aspect=('Square Frame', False),
    tachyon=('Ray Tracer', True)):
    xmin, xmax = sage_eval(xrange); ymin, ymax = sage_eval(yrange)
    P = plot3d(f, (x, xmin, xmax), (y, ymin, ymax), color=clr)
    html('<h1>Plot of $f(x,y) = %s$</h1>'%latex(f))
    aspect_ratio = [1,1,1] if square_aspect else [1,1,1/2]
    show(P.rotate((0,0,1), -zrot).rotate((1,0,0),xrot),
         viewer='tachyon' if tachyon else 'jmol',
         figsize=6, zoom=zoom, frame=False,
         frame_aspect_ratio=aspect_ratio)
}}}


attachment:tachyonplot3d.png

[[Anchor(eggpaint)]]

=== Somewhat Silly Egg Painter ===
by Marshall Hampton (refereed by William Stein)
{{{
var('s,t')
g(s) = ((0.57496*sqrt(121 - 16.0*s^2))/sqrt(10.+ s))
def P(color, rng):
    return parametric_plot3d((cos(t)*g(s), sin(t)*g(s), s), (s,rng[0],rng[1]), (t,0,2*pi), plot_points = [150,150], rgbcolor=color, frame = False, opacity = 1)
colorlist = ['red','blue','red','blue']
@interact
def _(band_number = selector(range(1,5)), current_color = Color('red')):
    html('<h1 align=center>Egg Painter</h1>')
    colorlist[band_number-1] = current_color
    egg = sum([P(colorlist[i],[-2.75+5.5*(i/4),-2.75+5.5*(i+1)/4]) for i in range(4)])
    show(egg)
}}}
attachment:eggpaint.png

== Miscellaneous ==

== Profile a snippet of code ==
{{{
html('<h2>Profile the given input</h2>')
import cProfile; import profile
@interact
def _(cmd = ("Statement", '2 + 2'),
      do_preparse=("Preparse?", True), cprof =("cProfile?", False)):
    if do_preparse: cmd = preparse(cmd)
    print "<html>" # trick to avoid word wrap
    if cprof:
        cProfile.run(cmd)
    else:
        profile.run(cmd)
    print "</html>"
}}}
attachment:profile.png


=== Evaluate a bit of code in a given system ===

by William Stein (there is no way yet to make the text box big):
{{{
@interact
def _(system=selector([('sage0', 'Sage'), ('gp', 'PARI'), ('magma', 'Magma')]), code='2+2'):
    print globals()[system].eval(code)
}}}

attachment:evalsys.png

=== A Random Walk ===

by William Stein

{{{
html('<h1>A Random Walk</h1>')
vv = []; nn = 0
@interact
def foo(pts = checkbox(True, "Show points"),
        refresh = checkbox(False, "New random walk every time"),
        steps = (50,(10..500))):
    # We cache the walk in the global variable vv, so that
    # checking or unchecking the points checkbox doesn't change
    # the random walk.
    html("<h2>%s steps</h2>"%steps)
    global vv
    if refresh or len(vv) == 0:
        s = 0; v = [(0,0)]
        for i in range(steps):
             s += random() - 0.5
             v.append((i, s))
        vv = v
    elif len(vv) != steps:
        # Add or subtract some points
        s = vv[-1][1]; j = len(vv)
        for i in range(steps - len(vv)):
            s += random() - 0.5
            vv.append((i+j,s))
        v = vv[:steps]
    else:
        v = vv
    L = line(v, rgbcolor='#4a8de2')
    if pts: L += points(v, pointsize=10, rgbcolor='red')
    show(L, xmin=0, figsize=[8,3])
}}}
attachment:randomwalk.png

=== 3D Random Walk ===
{{{
@interact
def rwalk3d(n=(50,1000), frame=True):
    pnt = [0,0,0]
    v = [copy(pnt)]
    for i in range(n):
        pnt[0] += random()-0.5
        pnt[1] += random()-0.5
        pnt[2] += random()-0.5
        v.append(copy(pnt))
    show(line3d(v,color='black'),aspect_ratio=[1,1,1],frame=frame)
}}}
attachment:randomwalk3d.png
{{attachment:taylor_series_animated.gif}}

Sage Interactions

This is a collection of pages demonstrating the use of the **interact** command in Sage. It should be easy to just scroll through and copy/paste examples into Sage notebooks. If you have suggestions on how to improve interact, add them here or email the sage-support mailing list. Of course, your own examples are also welcome!

Documentation links:

Examples:

Explanatory example: Taylor Series

This is the code and a mockup animation of the interact command. It defines a slider, seen on top, that can be dragged. Once dragged, it changes the value of the variable "order" and the whole block of code gets evaluated. This principle can be seen in various examples presented on the pages above!

taylor_series_animated.gif

interact (last edited 2021-08-23 15:58:42 by anewton)